Inspired by the MO question Homotopy extension of $E_\infty$-spaces.
Sending a map $f:S^1\to X$ to $f(\text{basepoint})$ gives a fibration $\Lambda X\to X$ with fiber $\Omega X$, where $\Lambda X$ is the free loop space of $X$ and $\Omega X$ is the based loop space of $X$.
On the other hand, a basepoint preserving map $e:X\to X$ gives a fibration $F_e\to X$ with fiber $\Omega X$ too, where $F_e$ is the homotopy fiber of $e$, i. e. the space of pairs $(x,p)$ where $x\in X$ and $p:[0,1]\to X$ is a path with $p(0)$ the basepoint and $p(1)=e(x)$.
For which $X$ does there exist an $e$ with fibrewise equivalence between $F_e$ and $\Lambda X$?
(Note that if one replaces the free loop fibration with the path fibration $PX\to X$ the answer is easily "all $X$": in that case $e$ can be taken the identity map.)